Converse, Inverse, Contrapositive

Date: 09/08/2001 at 23:57:03
From: Hana
Subject: Geometry - Converse,inverse and comtrapositive.
For the directions it says "Write the converse, inverse, and
contrapositive of each conditional. Determine if the converse,
inverse, and contapositive are true or false. If false, give a
counterexample."
I can't seem to do these:
All squares are quadrilaterals.
If a ray bisects an angle, then the two angles formed are
congruent.
Vertical angles are congruent.
Thank you!

Date: 09/09/2001 at 18:30:24
From: Doctor Achilles
Subject: Re: Geometry - Converse,inverse and comtrapositive.
Hi Hana,
Thanks for writing to Dr. Math.
Some basics about logic: When you're talking about converse, inverse,
and contrapositive, then you're dealing with sentences of the form
"if p, then q"
The converse of any such sentence is
"if q, then p"
The inverse is
"if not p, then not q"
And the contrapositive is
"if not q, then not p"
The problem with your questions are that they don't neatly fit into
the "if p, then q" format, so you need to first find EQUIVALENT
sentences that are "if p, then q."
Your first example says "all squares are quadrilaterals." That is the
same as saying "if x is a square, then x is a quadrilateral."
Your second example says "if a ray bisects an angle, then the two
angles formed are congruent." That is the same as saying "if a ray
bisects an angle, then the ray that is drawn within an angle forms two
congruent angles."
Your third example is just like your first example.
I hope this all gets you going. If you're still stuck, please write
back and I'd love to help you more.
- Doctor Achilles, The Math Forum
http://mathforum.org/dr.math/